Specializations of one-parameter families of polynomials

2004/05/07 by Farshid Hajir, Hajir, Farshid, Siman Wong +1
Computer Science · Mathematics · #12H25 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT #msc:12H25

paper · pdf · doi:10.48550/arxiv.math/0405139

arxiv created 2004/05/07 · openalex publication_date 2004/05/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a number field, and let lambda(x,t)∈ K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of alpha∈ K for which the specialized polynomial lambda(x,alpha) is K-reducible. We apply this to show that for any fixed n>=10 and for any number field K, all but finitely many K-specializations of the degree n generalized Laguerre polynomial are K-irreducible and have Galois group S_n. In conjunction with the theory of complex multiplication, we also show that for any K and for any n>=53, all but finitely many of the K-specializations of the modular equation Phi_n(x, t) are K-irreducible and have Galois group containing PSL₂(Z/n).

Related

• Families of polynomials and their specializations
• On the Galois group of Generalized Laguerre Polynomials
• On parametric and generic polynomials with one parameter
• Explicit Hilbert Irreducibility
• Effective Hilbert's Irreducibility Theorem for global fields